ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Death is not automatically a missing outcome

Example

Suppose the scientific target is a quality-of-life score at time $t$ for all participants enrolled at baseline, with positive probability of death before $t$ , and the score is defined only for people alive at $t$ . If death before $t$ makes the score itself undefined, those post-death values are not merely unrecorded full-data components. An unconditional mean over all enrolled participants has no value until the scientist specifies a different target, such as a composite outcome or a survivor-specific target.

Facts & Assumptions

Given: A population probability space of all baseline-enrolled participants with $P (D = 1) > 0$ ; $D = 1$ means death before $t$ ; and the proposed score $Q_{t}$ is defined only when $D = 0$ , with no post-death value stipulated.

[F1]

Random elements and real random variables: a real random variable is a measurable map from the entire probability space to $R$ .

[F2]

Full and observed data under a missingness pattern: the full-data vector contains the values that define the target, while the response pattern records which of those values are available.

[F3]

Estimands on a statistical model: an estimand is a specified function on the model's parameter set.

Verification

1.1F1given

Since $Q_{t}$ has no value stipulated on the event ${D = 1}$ , it is only a function on ${D = 0}$ and is not a map from the whole baseline probability space to $R$ . It therefore is not a real random variable on the stated population by [F1].

2.1F2F3step 1.1given∎

The observed-pattern construction [F2] can mask a coordinate of a full-data vector, but it does not define a value where the scientific outcome itself does not exist. Thus $E (Q_{t})$ over all enrolled participants is not a well-defined target function as required by [F3]. Assigning a post-death value, using a composite outcome, or restricting to survivors defines a different scientific target and must be stated separately.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Hernán and Robins, Causal Inference: What If, Chapter 8, §§8.4–8.5, and §12.6 (standard reference, not scraped)
National Research Council, The Prevention and Treatment of Missing Data in Clinical Trials, Chapter 4, pp. 47–49 (standard reference, not scraped)
Lu Mao, Missing Data: Foundations and Methods, Chapter 1, §1.1 (standard reference, not scraped)